| Week of | Aug 23
| Sep 6
| Oct 2
| Nov 6
| Phy 205 page |
| Aug 28 | Sep 11 | Oct 9 | Nov 13 | ||
| Sep 18 | Oct 16 | Nov 20 | Notes for Fall, 1999 | ||
| Sep 25 | Oct 23 | Nov 27 | |||
| Oct 30 |
28 Aug
Vectors and more vectors. How do you add them? (Two ways to describe
it)
How do you subtract them? (Three ways to describe this)
How do you multiply them by scalars? (Only one way)
How do you multiply them by each other? (Didn't do this yet, but there
are two ways that we will use)
How do you divide a vector by another vector? (You DON'T)
How do you conveniently manipulate them? (Graphically and using
components. I'll do the latter Wednesday.)
What good are vectors? Examples of physical ideas described by vectors include velocity, acceleration, torque, gravitational field, magnetic field, and many more.
30 Aug To compute with vectors, it is convenient to introduce some auxiliary vectors, sort of like using a coordinate system to do graphs. These "basis vectors" are denoted as i and j. (Not all browsers have the "hat" symbol built in, so I'm using bold face type here.) You can write any vector in the plane as a sum of two vectors -- one pointing in the direction of i (i-hat) and the other pointing in the direction of j (j-hat). In three dimensions, of course you'd need a third, call k (k-hat).
In terms of these, it's easy to do computations involving the sums and differences of vectors; all you need do is add or subtract the respective components. The magnitude of the vector is easy to get too; just use the Pythagorean theorem.
As a demonstration of how to use these vectors, use a force table that's set up so that it tips over if it is out of balance. Unless you compute the components of the vectors properly, it won't stay up.
1 Sept Some more examples of using components to compute with vectors. Take the example of the vectors X = 4mi + 3mj and Y = 5mi + 12mj. Draw both of these.
Compute their sum and difference, X + Y and X - Y. Also draw the pictures and see if the picture conforms to the arithmetic.
Compute the magnitudes of X and Y. I picked the components so that this comes out to be simple. Then find the magnitudes of the sum and difference that we just found. In this case, I used it as an excuse to show some misteaks that I've seen people make in previous classes.
There are various ways to define the product of two vectors. We will use only two of them, (1) the "scalar product" or "dot product" (2) the "vector product" or "cross product." The notations for these are respectively A·B and AxB. Simply putting the two vectors next to each other as you might do in algebra to multiply numbers doesn't tell you which of these two products you mean. Besides, XY actually does have a meaning as still a third type of product. (You don't want to know.)
The dot product of X and Y is defined to be X Y cos q, where q is the angle between the two vectors. Examine a few special cases of this product to see what it does. If the vectors are along a straight line (either parallel or anti-parallel), this behaves like the multiplication of ordinary numbers. If the vectors are at right angles to each other, the result is very different; it is zero. The product of two non-zero vectors can be zero. This product will make its appearance when we get to the subject of work and energy.
